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Source. Alain Plagne, Recent progress on finite $B_h[g]$ sets, author's manuscript (no venue or year printed), Section 3 (pp. 8-13), formula (13) on p. 8, and Section 3.1 (pp. 9-10), Problem 4 on p. 9, as identified on the source card. The file prints no page numbers; pages are counted from its first page.

Statement

Setting. $F_{2,1}(N)$ is the largest size of a Sidon set ($B_2[1]$ set: all sums $a+b$ with $a\le b$ distinct) contained in $\{1,\ldots,N\}$. The paper recalls (p. 8, formula (13)), as Lindström's form of the Erdős-Turán result,

$$F_{2,1}(N)\le\sqrt N+N^{1/4}+1.\qquad(13)$$

Problem 4 (p. 9), which the paper presents as a problem of Erdős. It asks, in order:

1. whether (13) can be improved asymptotically;
2. whether $F_{2,1}(N)\le\sqrt N+O(1)$, which it labels Erdős's conjecture;
3. or whether $F_{2,1}(N)\le\sqrt N+O(N^\varepsilon)$ for every $\varepsilon>0$;
4. more modestly, whether the exponent $1/4$ in (13) can be improved;
5. at least, with

$$\lambda=\limsup_{N\to+\infty}\frac{F_{2,1}(N)-\sqrt N}{N^{1/4}},$$

whether the bound $\lambda\le1$, which (13) gives, can be improved. The printed question reads "can one improve one $\lambda\le1$?" [sic].

After the problem the paper says $\lambda<1$ can probably be achieved and asks about proving $\lambda=0$, if true (p. 9).

Read depth. Claims checked: formula (13) and the five questions were read clause by clause on pp. 8-9. The problem is stated as open; there is no proof to check.

Proof pointer

None: an open problem. The paper sketches on p. 9 why counting differences instead of sums improves the trivial bound $F_{2,1}(N)\lesssim2N^{1/2}$ (formula (14)) to $\sqrt2N^{1/2}$, the idea behind (13).

Dependencies

Formula (13), cited to B. Lindström, An inequality for $B_2$ sequences, J. Combin. Theory 6 (1969), 211-212.

Bears on

• Problem 30: the problem asks whether $h(N)=N^{1/2}+O_\epsilon(N^\epsilon)$ for every $\epsilon>0$, where $h(N)$ is the paper's $F_{2,1}(N)$. Question 3 of Problem 4 is the upper half of that statement; a yes to question 2 would give it too. Neither addresses the lower half, $h(N)\ge N^{1/2}-O_\epsilon(N^\epsilon)$, which Problem 30 also requires. The paper proves nothing on either.